The fabrication of a microstructured optical fiber, or MOF, typically involves the drawing down of a preform comprising a cylinder of glass or polymer 1-3 centimeters in diameter containing a pattern of axial channels running through its length, usually 10-30 centimeters.
The preform is held at one end in a movable clamp at the top of a draw tower and pushed downwards at a specified feed speed through a heated zone in which the glass is heated and softens so that the heated preform can be pulled from below at a draw speed significantly larger than the feed speed. The ratio of the draw speed to the feed speed is known as the draw ratio. Both the draw ratio and the temperature are important control parameters in the draw process. The draw ratio determines the reduction in the area of the cross-section as it travels along the “neck-down length” of several centimeters (typically of the same order as the heated zone length).
To obtain a fiber having a typical diameter of 100-200 micrometers from a preform with a diameter of a centimeter, the draw ratio will typically be in excess of 4000. The temperature determines the material viscosity and, in turn, the fiber tension, which must be within an appropriate range—too small and its diameter will be difficult to control; too large and the fiber will break. At the base of the draw tower the (now cool) fiber is wound around a rotating drum so that it can be conveniently stored for future use. The draw process through the neck-down length is depicted in FIG. 1.
For microstructured optical fibers of interest in applications, there is a large variation in the number of channels, from just a few to perhaps 100 or more. The number of channels is not necessarily large, thereby rendering the proposition of a mean-field model of the cross-plane structure unlikely to be useful in practice. Moreover, it is often the unwanted deformations of specific channels in the cross-plane geometry that leads to compromised optical properties of the fiber.
Models that account for the different shape evolutions of all the individual channels are therefore highly desirable.
Previous approaches have involved direct numerical simulations of fiber draws. However, it is desirable to avoid the computational cost and necessity of full numerical simulation. Also, direct numerical simulations of fiber draws are of little practical use in the solution of the “inverse problem” where it is required to determine the initial geometry of preform configuration (such as the size and shape of the outer boundary and the size, shape and position of preform holes) and the set of manufacturing draw parameters (such as draw ratio, draw tension, and pressure, for example) that will, at the end of the draw, lead to the desired end-state geometry for the fiber.
With a view to finding a fast and accurate model of the forward problem, and providing a regularization mechanism for the challenging inverse problem, the inventors proposed a reduced model, called the generalized “elliptical pore model”, or EPM, to facilitate fast and accurate simulations of the shapes of multi-channel microstructured optical fibers during the fabrication draw process as described in the paper by P. Buchak, D. G. Crowdy, Y. M. Stokes & H. Ebendorff-Heidepriem, J. Fluid Mech., (2015); incorporated herein by reference. To provide a complete description of the fiber drawing process and to calculate relevant draw parameters, this model was coupled to a description of the axial flow.
The EPM was originally proposed in a different context to the MOF application described here, in a paper by D. G. Crowdy, “An elliptical-pore model of late-stage planar viscous sintering”, J. Fluid Mech., 501, 251-277, (2004). In this paper, the EPM was used to approximate the evolution of a doubly periodic square (four pores) and hexagonal (six pores) arrays of pores shrinking under the effects of surface tension and it was found to give excellent agreement with full boundary integral simulations of that problem performed by C. Pozrikidis, in “Expansion of a compressible gas bubble in Stokes flow”, J. Fluid Mech., 442, 171-189, (2001) and Pozrikidis, “Computation of the pressure inside bubbles and pores in Stokes flow”, J. Fluid Mech., 474, 319-337, (2003).
To describe the evolution of the cross section of a MOF, the EPM consists of a reduction of the solution to the full free boundary problem to the solution of a set of ordinary differential equations. It is suited to MOFs with a large number of channels and, for a wide class of geometries, it advantageously obviates the need for full numerical simulations; this has been extensively confirmed by direct comparison of the EPM predictions against the results of such simulations.
In the elliptical pore model, the basic idea is to resolve the evolution of each “pore” (representing the cross section of a channel) under the assumption that it remains elliptical as it evolves in a local linear flow induced by all the other pores. In the “far-field” of each pore all other pores are modelled as a point stresslet and point source/sink combination situated at its geometrical centroid. The outer boundary is assumed to remain circular as it evolves under the influence of the contained distribution of point singularities.
In the EPM, each channel is described mathematically by a time-evolving conformal map from the interior of the unit disc |ζ|=1. Each such mapping has the form
                                                        z              n                        ⁡                          (                              ζ                ,                τ                            )                                =                                                    ℨ                n                            ⁡                              (                τ                )                                      +                                                            α                  n                                ⁡                                  (                  τ                  )                                            ζ                        +                                                            β                  n                                ⁡                                  (                  τ                  )                                            ⁢              ζ                                      ,                            (        1        )            where τ is the time variable, Zn is the centroid position and the parameters αn(τ)∈R and βn(τ)∈C encode the orientation, area, and eccentricity of the elliptical hole (it may be shown, for example by substituting ζ=eiφ into (1), that the unit ζ-circle is transplanted under this mapping to the boundary of an ellipse). The parameter αn(τ) can be taken, by the freedoms of the Riemann mapping theorem, to be real without loss of generality.
However, there remains a need to improve the far-field approximation of the effect of each pore on the others. The possibility of a higher-order generalization of the EPM, referred to as a generalized pore model or GPM in the sequel, was discussed and explored at the same time as the EPM was proposed in the original paper by Crowdy (2004).
In this document, the terms “preform geometry” and “fiber geometry” refer to the size of the preform and fiber, respectively, as well as the locations, sizes, and shapes of the channels in the cross-section. The term “drawing process parameters”, or, more simply, “draw parameters”, refers to the parameters associated with the fiber drawing process. Examples of such parameters are (1) the “feed speed”, the speed at which the preform is fed into the heated zone, (2) the “draw speed”, the speed at which the fiber is wound around the drum, (3) the “draw tension”, the force used to draw the fiber around the drum, and (4) the pressures applied to the air inside the channels. The draw parameters may also include properties of the preform being used, such as its diameter and the surface tension coefficient for that material. Depending on the situation, some parameters may be specified while others may need to be calculated.